Calegari–Mazur conjecture on infinitesimally classical deformations

Let $F$ be an imaginary quadratic field and let $p$ be a split prime. Suppose that $\rho:\operatorname{Gal}(\overline{F}/F)\to\operatorname{GL}_2(E)$ is continuous, irreducible, nearly ordinary, unramified outside a finite set of places, and admits infinitesimally classical deformations. Calegari–Mazur conjecture. One of the following holds: there exists a character $\chi$ such that $\chi\otimes\rho$ descends to a representation over $\mathbb{Q}$; or the projective image of $\rho$ is dihedral, and the determinant character descends to $\mathbb{Q}$ with a CM fixed field in which a prime above $p$ splits. This conjecture gives the expected exceptional mechanisms for such deformations, namely base change or complex multiplication; its status is not resolved in the supplied text.